Elliptic functions lecture 4. The sigma function
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- Опубликовано: 30 янв 2025
- This lecture is part of a series of lectures on elliptic functions.
It covers the sigma function, the simplest theta function, and discusses line bundles over C/L.
For the other lectures in the course see
• Elliptic functions
this series is what I was missing in my life
Such a great lecture series.
Thank you Richard much appreciated
at 0:30 you say that the Weierstrauss p function has 2 poles on the fundamental domain and the Jacobi functions have 1 pole on the fundamental domain which seems to be backwards
edit: I realized my mistake. In the last video, we modified the fundamental domain but changed the our condition on the periodicity so that f(z+lambda)=\pm f(z).
edit 2: I imagine that by saying the Weierstrauss p function has 2 poles, we really mean that it has one double pole and that this forces the function to have two zeroes in the fundamental domain by the argument principle
Keep up the good work Richard. Take a look at mixed boundary value problem with electrostatic disk with hole/s. I try to contribute as much as I can we can narrow this down my friend :)
Extremely interesting content as always! Thank you for makin these series.
My first thoughs when trying to create a periodic function thetha(z) with zeros at the lattice, with the Weierstrass P function with poles of order 2, would have been to start with thetha(z) ~1/sqrt(P(z)). In the video is follows that P(z) = -log(thehta(z)) ' ', whereas in my case it would be P(z) ~ 1/thetha(z)^2. I understand that the second identiy is wrong because it makes P(z) no longer periodic since thehta(z) is not exactly periodic, but I feel maybe trying to correct that discrepancy, instead of doing the integral twice, could be an alternative deffinition?
Could someone tell me where my assumptinos went wrong?
Edit:
After watching the rest of the video, I now understand that the P(z) should in fact be expressable by a ratio of products of thetha. Somthing like P~ thehta(z-zero1) thehta(z-zero2) /theha(z)^2 in the fundamental domain. By Liouvile's theorem, there should be a constant of proportionality between these two functions. Therefore you have an implicit definition of thetha(z) as function of P(z)? I wonder if it is easy to invert...
Edit2:
I continued down this rabit hole, and started considering combining what we know. For simplicity I call thetha = T
Since P(z) = - log(T(z)) ' ' , Therefore using the second definiton, withouth knowing the constant
log(P(z)) = log(T(z-zero1)) + log(T(z-zero2)) - 2 log(T(z)) + Constant, Differentiating twice
log(P(z)) ' ' = log(T(z-zero1))' ' + log(T(z-zero2)) ' ' - 2 log(T(z)) ' ', Using the first identity
log(P(z)) ' ' = - P(z-zero1) - P(z-zero2) + 2 P(z)
This is an interesting functional equation however I have not been able to find anything relatd to it, has anyone seen a similar identity before? Is it really an Identity? I am aware that importnt constants have been ommited and I might have been sloppy with my algebra, so I might be wrong of course, but it is still fun to think about it!
Hello, Can i use sigma function to prove the infinite product of lemniscate sin ? Thank you !
This covers lots of gaps from "easy pre-requisite" and introductions into elliptic curves cryptography.
If you have a 1-cocyle c_\lambda(z), is it guaranteed that there is a function f(z) satisfying f(z+\lambda) = c_\lambda(z) f(z)? (It is given that c_{\lambda+\mu}(z) = c_\lambda(z) c_\mu(z+\lambda), so it is tempting to sum over \mu, but that seems very unlikely to converge.)
Yes it's theta
Laplace beltrami
Oh dear. It has gone very tricky to understand.
ohio sigma rizz
S I G M A
The LIGMA function